论文标题
一维随机几何形状的盒子计数尺寸
Box-counting dimension in one-dimensional random geometry of multiplicative cascades
论文作者
论文摘要
我们在随机乘法级联函数$ f $下研究了集合$ e \ subset \ mathbb {r} $的图像的盒子计数维度。 Benjamini和Schramm在随机几何形状的背景下建立了Hausdorff维度的相应结果，并且对于足够的规则集，相同的公式适用于盒子计数尺寸。但是，我们表明这一般远非如此，并且我们明确地计算出一种非常不同的性质的公式，该公式几乎可以确保在set $ e $组成收敛序列时，几乎可以确定随机图像$ f（e）$的盒子计数尺寸。尤其是，$ f（e）$的盒子计数尺寸比仅在其尺寸上更巧妙地取决于$ e $。我们还为一般集合$ e $的随机图像的盒子计数尺寸获得了下限和上限。
We investigate the box-counting dimension of the image of a set $E \subset \mathbb{R}$ under a random multiplicative cascade function $f$. The corresponding result for Hausdorff dimension was established by Benjamini and Schramm in the context of random geometry, and for sufficiently regular sets, the same formula holds for the box-counting dimension. However, we show that this is far from true in general, and we compute explicitly a formula of a very different nature that gives the almost sure box-counting dimension of the random image $f(E)$ when the set $E$ comprises a convergent sequence. In particular, the box-counting dimension of $f(E)$ depends more subtly on $E$ than just on its dimensions. We also obtain lower and upper bounds for the box-counting dimension of the random images for general sets $E$.
